Delft University of Technology
Power in sports
A literature review on the application, assumptions, and terminology of mechanical power
in sport research
van der Kruk, Eline; van der Helm, Frans; Veeger, DirkJan; Schwab, Arend DOI
10.1016/j.jbiomech.2018.08.031
Publication date 2018
Document Version
Accepted author manuscript Published in
Journal of Biomechanics
Citation (APA)
van der Kruk, E., van der Helm, F., Veeger, D., & Schwab, A. (2018). Power in sports: A literature review on the application, assumptions, and terminology of mechanical power in sport research. Journal of
Biomechanics, 79, 1-14. https://doi.org/10.1016/j.jbiomech.2018.08.031 Important note
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1 2 3 4
POWER IN SPORTS: A LITERATURE REVIEW ON THE
APPLICATION,
ASSUMPTIONS,
AND
TERMINOLOGY
OF
MECHANICAL POWER IN SPORT RESEARCH.
E. van der Kruk1, F.C.T. van der Helm1, H.E.J. Veeger1and A.L. Schwab1 ,1Department of Biomechanical Engineering, Delft University of Technology, Mekelweg 2, Delft, The Netherlands
Tel:+3115-2784270
e.vanderkruk@tudelft.nl
Key words: mechanical power, internal power, external power, mechanical energy expenditure,
joint power
5
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Power in Sports: a literature review on the application, assumptions, and
7terminology of mechanical power in sport research.
8E. van der Kruk1, F.C.T. van der Helm1, H.E.J. Veeger1 & A.L. Schwab1 9
Abstract
10
The quantification of mechanical power can provide valuable insight into athlete performance
11
because it is the mechanical principle of the rate at which the athlete does work or transfers energy to
12
complete a movement task. Estimates of power are usually limited by the capabilities of
13
measurement systems, resulting in the use of simplified power models. This review provides a
14
systematic overview of the studies on mechanical power in sports, discussing the application and
15
estimation of mechanical power, the consequences of simplifications, and the terminology. The
16
mechanical power balance consists of five parts, where joint power is equal to the sum of kinetic
17
power, gravitational power, environmental power, and frictional power . Structuring literature based
18
on these power components shows that simplifications in models are done on four levels, single vs
19
multibody models, instantaneous power (IN) versus change in energy (EN), the dimensions of a model
20
(1D, 2D, 3D), and neglecting parts of the mechanical power balance. Quantifying the consequences
21
of simplification of power models has only been done for running, and shows differences ranging from
22
10% up to 250% compared to joint power models. Furthermore, inconsistency and imprecision were
23
found in the determination of joint power, resulting from inverse dynamics methods, incorporation of
24
translational joint powers, partitioning in negative and positive work, and power flow between
25
segments. Most inconsistency in terminology was found in the definition and application of ‘external’
26
and ‘internal’ work and power. Sport research would benefit from structuring the research on
27
mechanical power in sports and quantifying the result of simplifications in mechanical power
28
estimations.
3 30
1. Introduction
31
Mechanical power is a metric often used by sport scientists, athletes, and coaches for research and 32
training purposes. Mechanical power is the mechanical principle of the rate at which the athlete does 33
work or transfers energy to complete a movement task. A mechanical power balance analysis can 34
provide valuable insight in the capability of athletes to generate power, and also in technique factors 35
affecting the effective use of power for performance. The estimates of mechanical power are usually 36
limited by the capabilities of motion capture systems, resulting in the necessity to use simplified 37
power models. However, due to the introduction of these simplified models and thus variation in 38
how power is calculated, the overview in literature in the terminology and estimation of mechanical 39
power is disordered. Furthermore, the validity of the simplifications is often disregarded. 40
The inconsistency in the use and definition of power came to our attention, when attempting to 41
estimate the mechanical power balance in speed skating (Winter et al. 2016; van der Kruk 2018). 42
Although thorough reviews exist addressing the issues of the mechanical power equations (van Ingen 43
Schenau & Cavanagh 1990; Aleshinsky 1986) and mechanical efficiency (van Ingen Schenau & 44
Cavanagh 1990), we found inconsistencies in the (post 1990) literature on the power estimations 45
and terminology. Moreover, the quantification on consequences of simplifications has usually been 46
disregarded. This not only makes the choice for a proper power model complicated, but also 47
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hampers interpretation and comparison to the literature. Providing insight into the interrelations 48
between the different models, estimations, and assumptions can benefit the interpretation of power 49
results and assist scientists in performing power estimations which are appropriate for their specific 50
applications. 51
The aim of this study is to provide an overview of the existing papers on mechanical power in sports, 52
discussing its application and estimation, consequences of simplifications, and terminology. 53
54 55
2. Method
56
A literature search was carried out in July 2017 in the database Scopus. The keywords “mechanical 57
power” and “sport” were used in the search (128 articles) (Search 1). The search was limited to 58
papers in English. Abstracts of the retrieved papers were read to verify whether the article was 59
suited to the aim of the paper, papers that estimated ‘power’ for a sporting exercise were included 60
(resulting in 94 articles). Three additional searches were performed in August 2017 addressing three 61
specific power estimations, combining the keyword “sport” with “external power” (30 articles) 62
(Search 2), “internal power” (4 articles) (Search 3), and “joint power” (35 articles) (Search 4), 63
restricted to articles published after 1990. Again, the abstracts of the retrieved papers were read to 64
verify whether the papers were suited for the current review. Papers that estimated ‘power’ for a 65
sporting exercise were included (resulting in respectively 13, 3, and 26 articles). 66
3. Application of the term power
67
When the terms mechanical power and sport were used in articles, the scope of the papers can 68
roughly be divided into two categories: the term power was either used as a strength characteristic 69
or performance measure (approximately 75% of the articles), or as an indication of mechanical
70
energy expenditure (MEE) (muscle work), which we focus on in this review.
71
The first application was mainly found in fitness and strength studies. Power is then wrongly used as 72
a strength measure, attributed to a certain athlete (Winter et al. 2016). This would implicate that 73
(peak) mechanical power is a synonym for short-term, high intensity neuromuscular performance 74
characteristic, which is directly related to performance of an athlete. However, as Knudson (2009) 75
also discusses, a peak power is not a fixed characteristic of a certain athlete. The power estimation in 76
a certain exercise, e.g. the well-known vertical jump (Bosco et al. 1983), cannot be directly translated 77
into performance of an athlete for other movements. Secondly, while strength is a force 78
measurement, power is a combination of force and velocity (Minetti 2002); these two are not 79
interchangeable. 80
Power can of course be used as an indication of performance during endurance sports. In cycling 81
practices, power meters (e.g., SRM systems, Schoberer Rad Messtechnik, Welldorf, Germany) are 82
widely accepted and used as an indication of the intensity of the training or race. Since a SRM system 83
determines power as the product of pedal force and rotational velocity of the sprocket, under the 84
same conditions (e.g. equal frictional and gravitational forces), the cyclist with the highest generated 85
power per body weight over time (work) will be fastest. This is, however, not applicable for every 86
sport. For example, power generated by a skater not only generates a forward motion (in line with 87
the rink), but also a lateral one (perpendicular to the rink). The result of this being that the skater 88
that generates the most power is not necessarily the fastest one finishing. Technique factors will 89
determine the effectiveness of the generated power towards propulsion. 90
This review focuses on the second purpose of power estimation: as indication of mechanical energy 91
expenditure (MEE). Power is the rate of doing work, the amount of energy transferred per unit time.
92
The relationship between mechanical power, muscle power and metabolic power is shown in Figure 93
1. Metabolic power can be measured by the rate of oxygen uptake, from which the energy 94
5
expenditure for the complete body in time is estimated. Mechanical power can be determined by 95
applying the laws of classic mechanics to the human body, and by modelling it as a linked segment 96
model consisting of several bodies (Aleshinsky 1986). Both metabolic power and mechanical power 97
estimates eventually aim to approach muscle power (either via the metabolic or via the mechanical 98
approach). Although muscle work is closely related to the MEE for the movement, mechanical power 99
and work are far from an exact estimation of muscle power and work and thus from MEE. 100
The disparity between mechanical power and muscle power can, next to measurement inaccuracies, 101
be attributed to physiological factors. In a mechanical approach, the part of the muscle power which 102
is degraded into heat or non-conservative frictional forces inside the body or in antagonistic co-103
contraction is not taken into account (Figure 1). Neither is the power against conservative forces 104
taken into account, such as tendon stretch, which in principle can be re-used (van Ingen Schenau & 105
Cavanagh 1990). 106
107
108
Figure 1 The power flow in human movement. Metabolic power and work are a chemical process, estimated by
109
for example measuring lactate or oxygen uptake (a). Energy distributes into muscle power, maintenance power
110
and entropy. Muscle power results in mechanical power (force times contraction velocity), except for
non-111
conservative power (e.g., power due to heat dissipation, non-conservative frictional forces inside the body, or
112
when muscles work against each other) and conservative power (e.g. power due to conservative forces, which in
113
principle can be re-used such as with tendon stretch). It is possible to convert the mechanical power into an
114
actual estimation of muscle power by the use of musculoskeletal models (II). The mechanical power balance
115
consists of joint power, which is generated by the human, which results in the kinetic power, which is the rate of
116
change of the kinetic energy, frictional power, due to e.g. air resistance, environmental power, which is induced
117
by external forces and moments, and gravitational power. The mechanical power can therefore be estimated
118
by the joint power alone, or by the combination of kinetic, frictional, environmental and gravitational power.
E-119
gross is the ratio between the expended work (metabolic work) and the performed work (mechanical work).
120 121
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Figure 2 Free body diagram of a rigid segment model of a human (adopted from van der Kruk et al. (2018)). The
123
human body is here divided into eight segments; the feet (f), the legs (e), the thighs (t) , the pelvis (p) and a HAT
124
(h), which are the head-arms-trunk. Note that HAT can only be appropriately grouped for certain sports
125
activities (such as ones that focus on lower extremity movement). In other activities, the HAT should be taken as
126
separate segments. The forces acting on the human are the ground reaction forces and the air frictional forces.
127
There are joint forces and moments acting at the Ankle (A), Knee (K), Hip (H) and Lumbosecral (L) joints.
128
Indicated are the Center of Mass (COM) of each segment, the Center of Pressure of the air friction (CP), where
129
the air frictional force acts upon, and the Center of pressure of the ground reaction force (COP).
130 131
7
4. Mechanical power equations
132
Before elaborating on the interpretation of mechanical power in the literature, we first set-up the 133
complete human mechanical power balance equations (based on the work of Aleshinsky (1986) and 134
van Ingen Schenau & Cavanagh (1990)), to expound the terminology used in this review. The 135
equations are based on the free body diagram shown in Figure 2. 136
The human is modelled as a chain of N linked rigid bodies (N ≥ 1), where each body is identified as a 137
segment with index
i
. We start by writing down the power balance of every segment and then add 138them to come to the power balance for the complete system. For a better understanding of the 139
system behaviour we distinguish between the joint power, which is the mechanical power generated 140
by the human at the joints; the frictional power losses; the kinetic power, which is the rate of change 141
of the kinetic energy; the gravitational power; and the environmental power, which is the mechanical 142
power from external applied forces and moments. We here use the term environmental power to 143
avoid confusion, since the term external power has been used to describe several different models 144
(e.g. the change in kinetic energy of the centre of mass (COM), as well as the power measured with a 145
power meter in cycling) (see section 5.2.1). Then, for one segment
i
we can determine these powers 146from the Newton-Euler equations of motion by multiplying them with the appropriate velocities. 147
Starting with the translational part, the Newton equation, we get for segment
i
., 148
F
j i,
F
G i,
F
e i,- F
f i,
v
im
i
a v
i i (1) 149In which
F
j i, are the joint forces,F
G i, are the gravitational forces,F
e i,are the external forces, and 150, f i
F
are the frictional forces working at the segment (e.g. air friction, ice friction).a
i andv
i are 151respectively the linear acceleration and velocity of the segment. We write the translational power 152 balance equation as 153 , , , , , , , , , , j tr i G tr i e tr i f tr i k tr i
P
P
P
P
P
(2) 154Where
P
j tr i, ,,
P
G tr i, ,,
P
f tr i, ,,
P
e tr i, , are respectively the translational joint power, the translational 155gravitational power, the translational frictional power, and the translational environmental power. 156
, , k tr i
P
is the translational kinetic power. 157For the rotational power we can take the Euler equation of motion, expressed in the global reference 158
system, and multiply by the angular velocities at the segment, to come to the rotational power 159 equation, as in 160
j i, e i, f i,
i
i i
i d I dt
M M - M (3) 161Where
M
j i, are the joint moments,M
e i,are the external moments,M
f i, are the frictional moments, 162and
ois the segment angular velocity. We write the power as 163, , , , , , , ,
j ro i e ro i f ro i k ro i
P
+
P
- P
P
(4)164
Next, we add up the rotational and translational segment powers of all segments. The constraint 165
forces in the joints have no contribution to the total power equation, since only relative rotation at 166
the joint between the two segments is assumed (linked segment model), and therefore will drop out 167
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of the equation. Joint forces can redistribute energy between segments and links, but not add energy 168
to the total body system (Aleshinsky 1986). Note however, that if an applied inverse kinematics 169
method allows for translations in the joint, as in Ojeda et al. (2016), or a six degree of freedom joint 170
is applied (e.g., as is possible in biomechanical modelling software such as OpenSim (Delp et al. 2007) 171
and Visual3D (C-Motion,Germantown,MD, USA)), joint forces do play a role and the constraint forces 172
should be accounted for in the power determination (see section 5.1.3). 173
The total power equations for the system, now written in terms of joint power, kinetic power, 174
frictional power, gravitational power, and environmental power are, 175
j k f G e
P P P P P (5)
176
177
In which we have the joint power (
P
j) which is directly calculated using the moments at the joint ( 178j
M
) and the rotational velocities around the joint (
j), as in 179
1 1 , 1 1 1 1 N N j i i i i j j i j P
M
M
(6) 180We find the gravitational power in equation 5, as in 181 1 N G i i i
P
m
v
g
(7) 182And the frictional power, which consists of translational power and rotational power, 183 , , 1 1 N N f i fr i i fr i i i
P
M
v F
(8) 184And the environmental power, which consists of translational power and rotational power, 185 , , 1 1 N N e i e i i e i i i
P
M
v F
(9) 186And the change of kinetic energy in the segments, 187
1 1 N N seg k i i i i i i i idE
d
P
I
m
dt
dt
a v
(10) 188In summary, the mechanical power balance consists of five parts, joint power, kinetic power, 189
gravitational power, environmental power and frictional power. Joint power is generated by the 190
human, and is the result of muscle power. This entails that for the most complete estimation of 191
mechanical (human) power either the joint power should be determined directly through 192
measurements of joint torques and angular velocity, or indirectly via the sum of frictional, kinetic, 193
environmental and gravitational power,
P P P
f,
k, ,
e andP
G (Figure 1). Usually, these powers are 194approximated depending on the available recording methods, and therefore sometimes not all terms 195
in the mechanical power balance are estimated resulting in a simplified model. 196
197
9
Instantaneous power (IN) versus change of energy (EN)
198
Power is the amount of energy per unit of time. In the literature there are, apart from the different 199
models, two different approaches to estimate power. First, what is referred to as instantaneous 200
power (IN). Instantaneous power is power at any instant of time, which can be calculated using the 201
power balance equation presented earlier (van Ingen Schenau & Cavanagh 1990). The second 202
approach is by determining the change of kinetic and gravitational energy of a system (EN) over a 203
larger time span, e.g. the cycle time, and divide this energy over the larger ∆t. We know that the 204
kinetic energy at time
t
is: 205 , , , , , , ,1
1
2
2
T T k i t i t i t i t i t i tE
m
v
v
(11) 206And the gravitational energy at time
t
: 207, , ,
g i t i t
E
m g y
(12)208
Note that EN only estimates average mechanical power, and does not give insight into the power 209
development, or peak powers. Also, oscillatory movements will result in a zero outcome with EN (e.g. 210
walking). 211
5. Power models in the literature
212
Based on the mechanical power equations, we sorted the literature of Search 1-3 concerning the 213
estimation of mechanical power as an indication of mechanical energy expenditure in Tables 1 & 2. 214
For each study the power model (
P P P P P
j, k, f, g, e), the estimation approach (IN, EN) and the 215dimensions (1D, 2D, 3D) are indicated. Results show that simplifications are done on three scales: 216
the number of bodies (single body vs multibody), the recorded data (kinematic versus kinetic data), 217
and the time interval (IN versus EN). The analysis on results for the literature of Search 4, are given 218
separately in Table 3, divided into articles for single joints versus multi-joints, and work versus power 219
results. 220
5.1 Simplifications of power models
221
5.1.1 Single body models
222
When an athlete is simplified to a single mass, the assumption is that this mass is located at the COM 223
of the full body. Constructing the mechanical power balance (eq. 5) for this single body system 224
results in an equation with one body left, the COM, which automatically neglects any relative 225
motions between the segments and the COM, and any power related to these motions. Although this 226
single body approach is used often (27 papers, see Table 1), estimation of the impact of this 227
simplification has only been performed in two studies, both on running (Arampatzis et al. 2000; 228
Martin et al. 1993) .
229
Arampatzis et al. (2000) (see also Table 1) compared four mechanical power models in over-ground 230
running at velocities ranging from 2.5-6.5 m/s. Their results showed that the mean mechanical power 231
estimated with the single body model, based on the change in potential and kinetic energy, is 32% 232
higher than the power of the 2D joint power estimation at 3.5m/s running speed. Martin et al. (1993) 233
determined the mechanical power in treadmill running with three methods (see Table 1). Based on 234
their results, a single body kinematic approach resulted in a 47% lower mechanical power estimation 235
compared to joint power, running at 3.35 m/s. Since the neglected frictional power (air friction) at 236
these running speeds is relatively small (<1% of joint power, based on Tam et al. (2012)), the 237
difference between joint power estimation and the kinematic approach for the single body 238
estimation is attributed to the neglected relative motions of the segments to the COM and the fact 239
Table 1 Structuring of the literature for single body models Indicated are the terminology, the power estimation, the dimensions of the model (1D, 2D, 3D) and whether the power is estimated instantaneously (Instantaneous power (IN)) or via the change in energy over a time span (EN). Inconsistent terminology and oversimplifications are indicated in the final column. Article Terminology Dimensions Pj Pk TRANS Pk ROT Pf Pg Pe IN/ EN Comments Applicable topics from this review Single Body models Running Yanagiya et al. (2003 ) Mechanical power 1 X IN velocity of the belt times the horizontal force on the handle bar Directional power (see 5.2.2) Fukunaga et al. (1981 ) (sprint) Forward power 2 X IN Pantoja et al. (2016 ) (sprint) Mechanical power 1 X X IN di Prampero et al. (2014 ) (sprint) Mechanical accelerating power 1 X X IN Minetti et al. (2011 ) (skyscraper) External power (internal power) 1 X EN Regression for internal power Internal and external work (see 5.2.1) Gaudino et al. (2013 ) (soccer) Mechanical power 1 X EN Directional power (see 5.2.2) Arampatzis et al. (200 0 ) Mechanical power 2 X IN +14% mean mechanical power a [compared to joint power in same experiment, Table 2 ] Oversimplified model (see 5.1.1) 2 X X E N +32% mean mechanical power b [compared to joint power in same experiment, Table 2 ] Martin et al. (1993 ) COM kinematics approach 2 X X E N 47% mean mechanical power c [compared to joint power in same experiment, Table 2 ] Bezodis et al. (2015 ) External power 1 X EN Cycling Telli et al. (2017 ) External power X IN Internal and external work (see 5.2.1) Additional External power 3 X X E N Internal and external power (see 5.2.1) Van Ingen Schenau et al. (1992 ) External power 1 X X E N Swimming Seifert et al. (2010 ) External power, Relative power, absolute power 1X IN Fdrag measured: the swimmers swam on the MAD-system, which allowed them to push off from fixed pads with each stroke These push-off pads were attached to a rod which was connected to a force transducer, enablin g direct measurement of push-off forces for each stroke. Assuming a constant mean swimming speed, the mean propelling force equals the mean drag force. Toussaint and Truijens (2005 ) 1 X X – Theoretical, not measured Toussaint and Beek (1992 ) 1 X X – Theoretical, not measured Rowing Hofmijster et al. (2008 ) External power 1 X IN Buckeridge et al. (2012 ) External power 1 X IN Integral of handle displacement-handle force curve divided by time. Hofmijster et al. (2009 ) Internal Power 1 X Internal and external power (see 5.2.1) Colloud et al. (2006 ) External mechanical power X IN Fhandle*vhandle-Fstretcher*vstrec hter Speed skating Houdijk et al. (2000a,b) External power 1 X EN About 20% of the joint power consists of Pk + P g based on van der Kruk et al. (2018 ) de Koning et al. (2005 ) Power output 1 X X E N de Koning et al. (1992 ) (sprint) External Power 1 X X E N
Wheelchair Mason et al. (2011 ) External Power Output 1 X EN Fdrag measured: The drag test setup consisted of a strain gauge force transducer, attached at the front of the treadmill to the front of the wheelchair. Participants were instructed to remain stationary while the treadmill was raised over a series of gradients at a constan t velocity Veeger et al. (1991 ) External power 1 X EN Fdrag measured: A cable was connected between the wheelchair (standing immobile on a sloped treadmill) and a force transducer mounted upon a frame at the front of the treadmill. Fdrag equalled the force needed to prevent the wheelchair from moving backward under influence of belt speed and slope effects. Kayaking Jackson (1995 ) 1 X X EN Theoretical, not measured Nakamura et al. (2004 ) Internal power Regression function Internal and external work (see 5.2.1) Sideway locomotion Yamashita et al. (2017 ) External power, vertical power, horizontal power, lateral power 2 X IN Internal and external work (see 5.2.1) Directional power (see 5.2.2) Bench press Jandacka and Uchytil (2011 ) (soccer) 1 X IN Vertical velocity of the COM x ground reaction force of the bench to the floor Oversimplified model (see 5.1.1) a Based on mean mechanical power of Table 2 a t 3.5 m/s in A. Arampat zis et al. (2000) : ((Metho d 1 -Method 4)/Method 4) * 100%. b Based on mean mechanical power of Table 2 a t 3.5 m/s in A. Arampat zis et al. (2000) : ((Metho d 2 -Method 4)/Method 4) * 100%. c Based on Martin et al. (1993) :( ( _ WEXCH in Table 2-TMP in Table 4)/(TMP in Table 4))) * 100%.
Table 3 Articles found with the search terms joint power and sport. The literature was divided into estimating power or work of a single joint (the research est imated the joint power of individual joints), and power and work of multiple joints (joint power was taken over multiple joints). Noted are the applied inverse dynamics technique with reference (N.M. = not mentioned). For the work est imation, the conversion from power to work is given and whether positive negative work are separated. Articles are sorted on year of publication. Joint power Power per joint Movement Inverse dynamics method Paquette et al. (2017 ) Running ‘‘Newtonian inverse dynamics” N.M. Middleton et al. (2016 ) Cricket ‘‘Standard inverse dynamics analysis” N.M. Barratt et al. (2016 ) Cycling Inverse dynamics method Elftman (1939 ) Pauli et al. (2016 ) Squats, jumps N.M. N.M. Van Lieshout et al. (2014 ) Exercises N.M. N.M. Creveaux et al. (2013 ) Tennis [Method is fully described in paper] n.a. Kuntze et al. (2010 ) Badminton N.M. N.M. Riley et al. (200 8 ) Running ‘‘Vicon plug-in-gait” Vicon Dumas and Cheze (2008 ) Gait ‘‘Inverse dynamics based on wrenches and quaternion s” Dumas et al. (200 4 ) Vanrenterghem et al. (2008 ) Jumping N.M. N.M. Schwameder et al. (2005 ) Walking ‘‘Standard 2D inverse dynamics routine” N.M. Rodacki and Fowler (2001 ) Exercise ‘‘Newtonian equations of motion ” N.M. Jacobs and van Ingen Schenau (1992 ) Sprint ‘‘Linked segment model” Elftman (1939 ) Energy per Joint Movement Inverse dynamics method Power to work Absolute Schache et al. (2011 ) Running ‘‘A standard inverse dynamics technique” Winter (2009 ) integral of joint power over time Not absolute (pos and neg work) Hamill et al. (2014 ) Running ‘‘Newton-Euler inverse dynamics approach” N.M. N.M. Not absolute (pos and neg work) Sorenson et al. (2010 ) Jump Inverse Dynamics Visual 3d Integral of joint power over time Not absolute (pos and neg work) Yeow et al. (2010; Yeow et al. 2009 ) Landing jump N.M. N.M. Integral of joint power over time Not absolute (pos and neg work) Power multiple joints Movement Inverse dynamics method Strutzenberger et al. (2014 ) Cycling ‘‘Sagittal plane invers e dynamics” Visual 3D Integral of the summed ankle, knee, and hip powers – Energy multiple joints Movement Inverse dynamics method Power to work Absolute Greene et al. (2013, 2009 ) Rowing Custom prog ram Winter (2009 ) Sum of the joint mechanical energy N.M. Attenborough et al. (2012 ) Rowing Inverse dynamics Winter (2009 ) Integration of the absolute value of the power time series curve for each joint Absolute per joint Lees et al. (2006 ) Jumping ‘‘Inverse dynamics using standard procedures” Miller and Nelson (1973 ), Winter (2009 ) Time integral per joint ‘‘Standard procedure’, de Koning and van Ingen Schenau (1994 ); sum of left and right limb; Not absolute (pos and neg work) Devita et al. (1992 ) Running ‘‘An inverse dynamics method” N.M. Resultant joint powers around hip, knee and ankle joint were summed at each time point. Not absolute (pos and neg work)
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that only measured kinematic data were used in the single body, which is expected to be less 240
accurate than the combination of measured force and kinematic data. The difference in results 241
between the two studies is surprising, since the mechanical equations, running speeds, and joint 242
power models (14 versus 15 segments, 2D, absolute per joint) are similar for both studies, while the 243
only difference was the treadmill versus over-ground condition. Unfortunately, Arampatzis et al. 244
(2000) do not discuss this difference. 245
It is clear that, although there is no consensus on whether a single body model under- or 246
overestimates the mechanical power in running (see also section 5.2.1), both studies show significant 247
differences between a single body model and a joint power model. Since this is the consequence of 248
disregarding the motions of the segments and kinematic measurement accuracy, validity will likely be 249
different for different movements. 250
Three studies were found that determined the mechanical power in locomotion with a single body 251
model by multiplication of an environmental force (e.g. the measured ground reaction forces) times 252
the velocity of the centre of mass of the complete body (Arampatzis et al. 2000; Yamashita et al. 253
2017; Jandacka & Uchytil 2011). Theory of this model lays in the simplification of an athlete to one 254
rigid body being propelled by a force. Therefore, the ground reaction force, which acts on the foot is 255
now shifted to the COM and assumed to cause the movement of the complete (rigid) body. However, 256
although a force can be replaced by a resultant force acting at the COM without changing the motion 257
of the system, the work of the system will divert from the actual work. For example, the ground 258
reaction force in running, acting on the foot, in principle hardly generates power, after all the foot 259
has close to zero velocity (Zelik et al. 2015). By assuming that the force acts on the COM of the 260
athlete, the force suddenly generates all power (and therefore work). So although mechanically, with 261
the rigid body assumption, the simplified model is in balance, the validity of modelling an athlete as a 262
point mass (single body) driven by the ground reaction force is highly doubtful. The results of such a 263
model should in no case be interpreted as an indication of muscle power/work or MEE, since the 264
relationship with actual joint power is lost by the oversimplification of an athlete. 265
For single body power estimations, both IN approaches (e.g. Pantoja et al. 2016; di Prampero et al. 266
2014; Seifert et al. 2010) and EN approaches (e.g. Minetti et al. 2011; Gaudino et al. 2013; Houdijk et 267
al. 2000) were found. An EN approach results in an average mechanical power estimate. 268
Consequently, there is no insight into the course of power during the motion cycle, e.g. peak power. 269
Also, oscillatory motions are averaged such that positive and negative power would negate each 270
other, which are tricky assumptions for several sports like running, cycling, swimming, etc. Van der 271
Kruk (2018) found that the kinetic and gravitational power related to these oscillatory motions in 272
speed skating (zig-zag motion of the skater over the straight), appeared to account for almost 20% of 273
the joint power. Therefore, assumptions on ignoring velocity fluctuations, or motions that do not 274
directly contribute in the forward motion, should be well validated. Especially when working with 275
top-athletes or highly technical sports, these components could be the key-factors in an athlete’s 276
performance, therefore IN models seem more appropriate than EN models for understanding 277
performance (Caldwell & Forrester 1992). 278
5.1.2 Multibody models
279
Using a multi-body approach is much more complex than the single body approach, since the motion 280
of the separate body parts needs to be measured. The benefit of this approach is that the power per 281
segment gives insight into the distribution of power over the body. In the kinematic approach, only 282
recorded kinematic data are used to indirectly estimate mechanical power: frictional power, kinetic 283
power and gravitational power (
P
f ,P
kandP
G). The main difference with the joint power estimation, 284is the absence of measured force data. Furthermore, in the kinematic approach frictional power is 285
neglected in running and walking studies, and gravitational power in cycling studies. 286
11
The studies by Arampatzis et al. (2000) and Martin et al. (1993), which were mentioned earlier, 287
enable the comparison of a kinematic multi-body approach, which resulted in respectively 10% more 288
mechanical power and 56% less mechanical power when compared to the joint power estimation (at 289
respectively 3.5 m/s and 3.35 m/s) (Table 2). Again, their results are contradictory and largely diverge 290
in magnitude. However, the results do stress the need of accurate kinematic measurements in the 291
models. The approaches in which both recorded kinematic and force data were used to estimate 292
MEE correlated better with the aerobic demand of the athletes than the kinematic data only 293
approaches (Martin et al. 1993). 294
5.1.3 Joint power
295
Since we found several inconsistencies in estimating joint power in the articles of Search 1-3 (see 296
Table 2), we performed a specific search for joint power (Search 4). Analysis of these studies lets us 297
identify two classes of differences in joint power estimation: the inverse dynamics method (including 298
the degrees of freedom of the joints) and the estimation of power to work (see Table 3). 299
Joint power estimation requires the determination of joint moments and forces via an inverse 300
dynamics method. Although several methods exist to estimate joint moments (e.g. Dumas et al.
301
2004; Kuo 1998; Elftman 1939), the bottom-up approach (Winter 2009; Elftman 1939; Miller & 302
Nelson 1973) is still the most applied method, and referred to as the ‘standard inverse dynamics 303
method’ or ‘Newton(-Euler) inverse dynamics approach’ without citing further reference. However, 304
since the bottom-up approach can leave large residuals at the trunk and the joint power is largely 305
influenced by the inverse dynamics method (up to 31% (van der Kruk 2018)), there should be more 306
attention towards this part of the power estimation. 307
308
Underlying the inverse dynamics is the choice for the kinematic model, where we mainly found 309
differences in the degrees of freedom of the joint (van der Kruk et al. 2018). If translation is allowed 310
in the joints, the joint forces suddenly generate power (see eq. 2). Application of 6 DOF joints, and 311
therefore incorporation of translational joint power is becoming more common, due to the ever 312
more detailed 3D human joint models (e.g. OpenSim, Visual3D). The effect of these forces on the 313
joint power, and whether the translations are not part of residuals of the choice in inverse kinematics 314
method, rather than a physiological phenomenon, falls outside of the scope of this review (Ojeda et 315
al. 2016; Zelik et al. 2015). However, we want to make the reader aware that differences do occur 316
and thereby influence the joint power estimations, where the increase in complexity will not 317
automatically imply improvement. 318
319
The second class of difference was found in the integration of joint power to work (as indication of 320
MEE). For the power in a single joint, a separation is made between negative and positive power. 321
Negative power occurs when the moment around the joint is opposite to the angular velocity of the 322
joint, which would denote braking (dissipation of energy). With only mono-articular muscles, this 323
would imply the production of eccentric power. However, bi-articular muscles can ‘transfer’ power to 324
adjacent joints. Converting power into work is done by taking the integral of the power curve over 325
time. In the literature, the division is made between positive work and negative work (Schache et al. 326
2011; Yeow et al. 2009; Hamill et al. 2014; Sorenson et al. 2010). This division is made since, from a 327
biomechanical perspective, it is assumed that for negative muscle work (or eccentric muscle 328
contraction) the metabolic cost is lower than for positive muscle power requiring concentric muscle 329
contraction. However, there is no general consensus on the exact magnitude of this difference. 330
Caldwell & Forrester (1992) even argue that the division into positive and negative work should be 331
rejected, since mechanical power is an indication of muscle power, not metabolic cost and thus 1 J of 332
negative power reflects 1 J of positive power. However, currently the general consensus is to 333
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12
separate negative from positive work; musculoskeletal simulations might shed light on the difference 334
in magnitude in the future. 335
336
For power estimation in multiple joints, the estimation of mechanical work (indication of MEE) 337
becomes more complicated due to the power flow between segments (and thus joints); bi-articular 338
muscles activations can induce both negative and positive power simultaneously around adjacent 339
joints (Van Ingen Schenau & Cavanagh 1990). When no power flow is assumed, the integral of the 340
absolute joint power per joint is taken and summed over the joints (Attenborough et al. 2012). If 341
power flow is assumed, the joint powers are first summed over the joints and then the integral over 342
time is taken, again allowing for the separation of negative and positive power (Lees et al. 2006; 343
Devita et al. 1992). What the best approach is, has yet to be determined. Hansen (2003) found in 344
cycling that the MEE was most accurately measured with a model that allowed for energy transfer 345
only between segments of the same limb. Articles that do not report the method for MEE estimation 346
are inappropriate for comparison (e.g. Greene et al. 2013), since the difference between the two 347
methods can go up to >2.5x the MEE (measured in running (Martin et al. 1993). Note that this power 348
flow issue not only accounts for the estimation of joint power over several joints, but also for power 349
transfer between segments in other kinematic multi-body models (Willems et al. 1995). 350 [Table 3] 351 352 5.2 Inconsistent terminology 353 354
Figure 3 Free body diagram of a two-link segment body
355
5.2.1 Internal and external work
356
The terms internal and external power and work are often used. However, these terms are ill-357
defined, terminology is inconsistent, and the actual purpose of separation is dubious. We will discuss 358
these issues by considering a simple 2D two-link model (Figure 3). The mechanical power equations 359
of this simple model can be divided into external powers and internal powers. We here employ the 360
definition of internal power as the energy changes of the segments, relative to the COM of the 361
complete body (Aleshinsky 1986). The power equation for this model can be divided as follows: 362
13
2 2 2 2 1 1 1 2 2 1 1 12
2
2
/ / / / , Ncom com i ci com ci com ci i
com
i
N
x x y y x y
o com d com o com d com o o com o o com o
i
M
x
y
m
x
y
I
dE
d
d
M g y
dt
dt
dt
F
x
F
x
F
y
F
y
F
x
F
y
M
M
(13) 363in which the parts in the blue boxes represent the external powers, and the parts in the green boxes 364
the internal powers. Note that the external force Fo acts at
o
, and:365
;
/ /
o com o com o com o com
x
x
x
y
y
y
(14)366
Although these equations show that the system energy can be presented as a sum of external and 367
internal power, the mechanical work is not equal to the sum of the ‘internal’ and ‘external’ work 368
(Zatsiorsky 1998; Aleshinsky 1986). Take into consideration that: 369
;
/ /
com o o com com o o com
x
x
x
y
y
y
(15)370
If we then determine mechanical work by taking the absolute integral of the power equations 371
separated into internal and external power, we obtain: 372
2 2
2 2
2 2 1 1 1 2 12
2
2
/ / / / / / T T Ncom com i ci com ci com ci i
com
i
T T
T
x x y y x y
o o d com o o d com o o com o o com
T x y o o com o o c
M
x
y
m
x
y
I
d
d
M g y
dt
dt
dt
dt
F
x
F
x
F
y
F
y
F
x
F
y
dt
F
x
F
y
2 1 1 2 2 1 1 1 1 , T N om o i TM
M
dt
373 (16) 374As mentioned by Aleshinsky in 1986, there are external forces (
F
o) inside the ‘internal’ work, 375therefore the internal and external work are not independent measures. Moreover, the absolute 376
values (due to positive and negative work) destroy the balance. Members of the expressions in the 377
internal and external work are powers which regularly fluctuate out of phase, thereby cancelling 378
each other out. By treating them as independent measures, the work doubles instead of cancelling 379
out, while in reality these powers do not cost any mechanical energy (e.g. pendulum motion). 380
Replacing an actual system of forces applied to a body by the resultant force and couple does not 381
change the body motion. It can change, however, the estimation of performed work. Therefore, the 382
power of the external forces as a hypothetical drag force, when assumed this acts at the COM, can be 383
seen separate from the internal power (there is no relative velocity between the point of application 384
of the force and the COM). However, ground reaction forces, or any other forces with a point of 385
application different from the COM will be part of both the ‘internal’ and ‘external’ work, and 386
therefore are not independent measures (see also section 5.1.1). 387
Despite the mechanical incorrectness of the separation of internal from external work, and the 388
discussion involving these measures (Zatsiorsky 1998; van Ingen Schenau 1998), more recent 389
publications still make this distinction (e.g. Minetti et al. 2011; Nakamura et al. 2004), raising the 390
question what the benefit is of separating the mechanical energy into internal and external energies 391
if the separation is mechanically incorrect? In cases where the whole mechanical power balance is 392
estimated, there seems no point in dividing the power into internal and external power or work. This 393
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14
separation has not given additional useful insight into human power performance in sports so far. 394
The only application of the separation could be when a single body model is used and therefore only 395
external power can be measured. The balance ratio between internal and external power can then 396
be used to provide insight into the consequences of the simplification. 397
Adding to the confusion of the interpretation of external and internal power, is the inconsistent use 398
of the terms. The use of the term ‘internal’ is logically diffuse, while it might refer to muscular or 399
metabolic work (Williams 1985). In this literature review, two articles were found that used the 400
internal power for estimations different from the definition given above, defining internal 401
mechanical power loss as the part of power absorbed by the muscles that is lost to heat (estimated 402
as fluctuations in kinetic energy of the back and forth moving of the rower on an ergometer) 403
(Hofmijster et al. 2009), or the total energy required to move segments (Neptune & Van Den Bogert 404
1997). However, more models and interpretations of internal power have been published, that all 405
largely (up to 3x) differ in power output estimation (Hansen et al. 2004). 406
Also the term external power is inconsistently used. Aleshinsky (1986) defined the term as the 407
change in energy of the COM of the athlete, and can therefore be seen as a single body model. The 408
origin of the term lies in the assumption that the human generates power only to overcome external 409
forces (e.g. air friction, ground friction). In speed skating (Houdijk et al. 2000; de Koning et al. 1992), 410
wheelchair sports (Veeger et al. 1991; Mason et al. 2011) and swimming (Seifert et al. 2010), the 411
term external power is used for the estimation of frictional power (Pf), assuming that, under constant 412
velocity, this is equal to the power generated by the human. In rowing (Hofmijster et al. 2008; 413
Buckeridge et al. 2012; Colloud et al. 2006) and cycling (Telli et al. 2017), where ergometers are 414
available, the term external power is used to describe the power output measured by the ergometer, 415
what we define as environmental power (Pe). Note however, that the power output measured with 416
an ergometer or a system such as SRM is not necessarily the same as the COM movement. If a cyclist 417
stops pedalling on an ergometer but moves her or his upper body up and down, there is a COM 418
movement (due to joint power), but there is no power measured at the ergometer (Pe) (the cyclist of 419
course does not have to stop pedalling for the same effect). In running and walking, where the 420
frictional power is only marginal and environmental power in principle is zero, the term external 421
power is used to describe the change in kinetic energy (Pk) (Bezodis et al. 2015) and/or gravitational 422
energy (Pg) (Minetti et al. 2011) of the COM, but also for an estimation done by multiplication of the 423
ground reaction forces times the COM velocity (see section 5.1.1 on the reliability of this model). 424
More interpretations of external power can be found in Table 1. 425
So even though the term external power is well known and frequently used, the estimation is not 426
straightforward and interrelations are not always clear. The terms internal and external power can, 427
however, be structuralized and classified by the mechanical power balance from section 3, as was 428
done in Table 1 and 2. We propose a standard in section 6. 429
5.2.2 Directional power
430
In the studies on running and walking, we found many power terms related to some sort of direction: 431
forward power, lateral power, etc. (see Table 1 and 2). Since power is a scalar, it is in principle 432
incorrect to give the power a certain direction, although of course the forces and velocities related to 433
power have a direction. The separation of the mechanical power equations into these different 434
directions is actually not beneficial. Take for example a situation where there is no environmental 435
power acting on the human e.g. walking; in that situation the power equation simplifies to: 436
2 2
2 2
1
1 2 2 1 1 12
2
2
/ / , N Ncom com i ci com ci com ci i
com i i
M
x
y
m
x
y
I
d
d
M g y
M
dt
dt
437 (17) 43815
Although the translational left side of this equation can be divided into terms related to a certain 439
translational direction, the eventual power production, on the right side of this equation, cannot be 440
separated into these directions. Separating the left side of the equation into directional terms, is 441
completely dependent on the chosen global frame; moreover, ‘vertical’ power can very easily be 442
translated into a ‘lateral power’ without adding power to the system, e.g. due to centrifugal forces. 443
5.3 E-gross
444
This review clearly showed that there arise large differences in mechanical power estimation based 445
on the choice for a model. These differences also impact research studies which estimate metabolic 446
power with gross efficiency calculations (e-gross), which is the ratio between the expended work 447
(metabolic work) and the performed work (mechanical work). E-gross is often determined in a lab, 448
using VO2-measurements, to convert mechanical work into energy expenditure. Main causes in the 449
differences among athletes and inaccuracies in measurement of e-gross have been ascribed to the 450
metabolic side of the equation. However, determination of the mechanical power with simplified 451
models influences the e-gross estimation evenly well. When only part of the mechanical power 452
balance is determined, for example with a single body model, the dependency of e-gross to the 453
relative movements of the segments is neglected (e.g. de Koning et al. (2005)). If an athlete would 454
then change movement coordination (technique) between the submaximal experiment (where e-455
gross is set) and the actual experiment, the change in segment motion is neglected in the mechanical 456
power and thus in the metabolic power estimation. Especially for technique dependent sports (e.g. 457
swimming, speed skating), this seems an important fact. 458
6. Discussion
459
This review provided an overview of the existing papers on mechanical power in sports, discussing 460
the application and the estimation of mechanical power, the consequences of simplifications, 461
mechanically inconsistent models, and the terminology on mechanical power. Structuring the 462
literature shows that simplifications in models are done on four levels: single vs multibody models, 463
instantaneous power (IN) versus change in energy (EN), the dimensions of a model (1D, 2D, 3D) and 464
neglecting parts of the mechanical power balance. Except for the difference between single versus 465
multibody models in running, no studies were found that quantified the consequences of simplifying 466
the mechanical power balance in sport. Furthermore, inconsistency was found in joint power 467
estimations between studies in the applied inverse dynamics methods, the incorporation of 468
translational joint power, and the integration of joint power to energy. Both the validation on 469
simplification of models and the lack of a general method for joint power or work are research areas 470
well worth investigating. 471
The terms internal power and external power/work are, apart from the discussion on the actual 472
usefulness of this power separation, confusing, since several meanings were attributed to the terms. 473
The interrelations between the different interpretations of external power have been discussed here. 474
Based on the above, we suggest that it might be more clear to use the terms from the mechanical 475
power balance: joint power (eq. 6), gravitational power (eq. 7), frictional power (eq. 8), 476
environmental power (eq. 9) and kinetic power (eq. 10) and not use the terms internal and external 477
power or work. In case the power due to motion of the COM and due to motion of the segments 478
relative to the COM are to be separated for measurement conveniences, we propose to work with 479
the term Peripheral Power for moving body segments relative to the COM (Zelik & Kuo 2012; Riddick 480
& Kuo 2016). Note however, that these should not be interpreted as separate energy measures 481
(mechanical work). The awareness that terms internal and external work/power are not self-evident 482
and therefore need explanation and interrelation to the mechanical power balance, will reduce the 483
possibility of errors and increase the comprehension for the reader. 484
To quote Winter et al. (2016): ‘ if sport and exercise science is to advance, it must uphold the 485
principles and practices of science.’ This review only revealed the tip of the iceberg of the studies 486
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concerned with estimating ‘power’ in sport (the search term power and sport results in 9,751 articles 487
(August 2017)), but illustrates clearly that the sport literature would benefit from structuring and 488
validating the research on (mechanical) power in sports. By structuring the existing literature, we 489
identified some obstacles that may hamper sport research from making headway in mechanical 490 power research. 491 7. Conclusions 492 493
Performance is not a direct translation of mechanical power. 494
Mechanical power is not a direct estimation of muscle power. Mechanical work is also no 495
direct measure of energy expenditure for movement. 496
Mechanical power is estimated via the joint power directly, or via the sum of kinetic, 497
frictional, gravitational and environmental power; all other estimations are simplifications. 498
Due to limitations in human motion capture in sports, simplified models are employed to 499
determine power. Simplifications in models are done on four levels: single vs multibody 500
models, instantaneous power (IN) versus change in energy (EN), the dimensions of a model 501
(1D, 2D, 3D) and neglecting parts of the mechanical power balance. 502
Single body models by definition neglect the relative motion of the separate body segments 503
to the COM of the body. The resulting underestimation in power, as an indication of muscle 504
power, is rarely determined in sports, whereas this part of power is an essential part of the 505
mechanical power balance in technique driven sports as e.g. speed skating, swimming or 506
skiing. 507
IN models are more appropriate than EN models for understanding performance of elite 508
athletes. EN automatically results in determination of average power and therefore 509
oscillatory movements are averaged such that positive and negative power would negate 510
each other. 511
Little attention is given to the chosen inverse dynamics technique to estimate joint moments 512
and forces, although its influence on joint power estimation is large (e.g. 31% in speed 513
skating). 514
When 6DOF joints are applied (e.g. OpenSim, Visual3D), joint forces not only distribute 515
energy, as in the classical 3DOF joint rotational models, but also allow for translational 516
power; Sport researchers should be aware of the differences between these joint power 517
estimations. 518
There is no consensus on how negative and positive work in a single joint should be summed. 519
On the same note, there is no standard on whether to allow for energy flow between joints. 520
The chosen approach is not always clear from the articles, although factors of 2.5x difference 521
between approaches have been found. 522
The terms external and internal power and work are inconsistent. The terms can easily be 523
replaced by the terms joint power, kinetic power, gravitational power, frictional power and 524
environmental power mentioned in the mechanical power balance of this review paper, 525
which will avoid future confusion. 526
Gross-efficiency (e-gross) is not constant within and between athletes. Apart from metabolic 527
causes, this can also be caused by the procedure of mechanical power determination. 528
529
Acknowledgements
530
This study was supported by NWO-STW 12870. 531
Conflict of interest: none.
